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Grantor: Czech Science Foundation (GAČR)

**Objectives**:

Hysteresis, i.e., nonlinear relations exhibiting a complicated input-output behavior in form of nested loops that cannot be described by functions or graphs, occurs in many fields of science, e.g., in ferromagnetism, micromagnetics, solid-solid phase transitions, and elastoplasticity. Hysteretic systems carry a memory of their former states, which renders their input-output mapping both nondifferentiable and nonlocal in time, so that conventional weak convergence techniques for solving evolution systems fail. Therefore, dynamical elastoplastic processes with hysteresis are found in the mathematical literature much less frequently than quasistatic ones, and a substantial progress in this direction is necessary. In a recent breakthrough, it was shown that the three-dimensional single-yield von Mises constitutive law leads, after a dimensional reduction to beams or plates, to a multi-yield Prandtl-Ishlinskii hysteresis operator. It is in fact quite natural that the lower dimensional observer does not see any sharp transition from the purely elastic to the purely plastic regime as in the von Mises model: if a plate is bent then small plasticized zones start forming first near the boundary and then propagate to the interior, which still preserves a partial elasticity. This gradual plasticizing is reflected by the Prandtl-Ishlinskii superposition of single-yield elements that are successively activated. This new groundbreaking theory will be expanded to more complex structures like Mindlin-Reissner plates, and curved rods and shells. Temperature and material fatigue effects will be included. A thermodynamically consistent theory of temperature and fatigue dependent Prandtl-Ishlinskii operators will be developed, along with efficient and reliable numerical methods. Questions of theoretical and numerical stability, and the long time behavior of the system of energy and momentum balance laws are central objectives.

Main investigator:

IM team members:

Chytilová Lucie |
Deasy Fergal |

Participating institutions:

Institute of Mathematics, Czech Academy of Sciences, **Coordinator**

Silesian University in Opava